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292 X EXISTENCE THEOREMS
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8. (a) Generalize GronwalFs lemma (10.5.1.3) to the inequalities
»(/)<9<0+ aw p #*)»c»)«&
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f VifrX*) * + fla(0 f V»(»M*) *, etc.,
JO «/0
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where 9?, fa, i/f2»#i» #2 are regulated functions ^0. (Use induction on the number of
integrals on the right-hand side.)
(b) Let K(/, s) be a continuously differentiable function ^0 defined in [0, c] x [0, c].
Suppose there are two regulated functions g, h, defined and ^=0 in [0, c], such that 9K(/, s)/dt < g(t)h(s). Show that the inequality
w(0 < <p(t) 4-1 K(f, s)w(s) ds
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for a regulated function w ^ 0 implies
w(0 ^ <pi(t) 4~ 61 (t) h(s)w($) ds
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where (pi and 61 are functions which one can explicitly compute when the functions
9, g and r(t) — K(t, t) are known (consider the function y(t) = I K(r, s)w(s) ds and
JO
majorize its derivative).
(c) Apply (a) or (b) to the inequality
w(t) ^ / 4- X2t e~**w(s) ds -f- w(s) ds,
JO JO
where A > 0.
9. Let w be a real function defined in an open interval I c R, and suppose w is the primitive
of a regulated function w' whose points of discontinuity are isolated in I, and such that in each of these points w>'(/4-)> w'(t—); suppose in addition that if E is the set of points of discontinuity of w>', the second derivative w" exists in I — E and w*(x) ^ w(x) in I - E.
(a) Show that if a, b are two points of I such that w(a) = w(b) = 0, then w(x) ^ 0
for a < x < b (use contradiction). Conclude that for any three points Xi<x<x2inl one has
wfe) sinhfe — x) 4- w(x2) sinh(;t — Xi)
sinhfe — Xi)
(consider the difference w (x) — u(x)9 where u is a solution of the equation
u"(x) — u(x) = 0 taking the same values as w at the points xj. and x2). |
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6. LINEAR DIFFERENTIAL EQUATIONS
The existence theorem (10.4.5) can be improved in special cases:
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(10.6.1) Let IcKbean open ball of center t0 and radius r. Let f be continuous
in I x E ifK = R, continuously differentiable in I x E ifK = C, and such that |
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